A circle has a sector with area $27\pi$ and central angle $120^\circ$. What is the area of the circle? ${81\pi}$ $\color{#9D38BD}{120^\circ}$ ${27\pi}$
Solution: The ratio between the sector's central angle $\theta$ and $360^\circ$ is equal to the ratio between the sector's area, $A_s$ , and the whole circle's area, $A_c$ $\dfrac{\theta}{360^\circ} = \dfrac{A_s}{A_c}$ $\dfrac{120^\circ}{360^\circ} = 27\pi \div A_c$ $\dfrac{1}{3} = 27\pi \div A_c$ $A_c \times \dfrac{1}{3} = 27\pi$ $A_c = 27\pi \times 3$ $A_c = 81\pi$